A measure of structural complexity

In other words, the model of a system (its structure) formulated above uses the language of set theory and binary relations. Because of this, the following includes the statements and observations we can be guided by when estimating the structural complexity of system in the Polyhedral Dynamics approach: 1) designed mathematical model and the results of its analysis form the basis for creation of structural complexity estimate; to a certain extent it corresponds to the conclusion that measure of complexity depends upon a particular language of representation. The model replaces the actual system, and it means that we are «digressing» from the real complexity of the real structure(-s) and trying to analyze its image and build a theoretical estimate, which is based on the information summarized in the model, 2) complex system includes many parts (components) and connections between them. These features cannot be regarded as unique when defining a complex system, but tructural properties, together with behavioral properties, primarily define complex systems. The presence of connections testifies to a pendency of parts and existence of the order that leads to the wholeness (integrity) of the system. So far as we are speaking about complexity of structure - in other words, complexity of the actual relations, which held the components together - we stress this singularity. In the other cases, for example, when studying the complexity of behavior, it can be unimportant, since the system with simple structure (simple interacting elements) may be characterized by complex behavior, 3) the number of parts in the system should not be considered directly as a measure of structural complexity because the term «parts» is not uniquely defined, and it depends upon the level of description. If an observer (researcher) disposes of formation about the system, it gives him the possibility of creating several mathematical models and of performing a more detailed study of the system depending on the elements (parts) and interconnections representation level, 4) a degree and nature of interaction (weights of connections) between elements of a system are not taken into consideration — very often such information can be inaccessible at the initial stages of analysis, 5) most probably, complexity should not be regarded in a context of simple numeric value, because in such case it irretrievably «hides» the essential structural information expressed in the mathematical model and «amounts» distinctive multidimensional system’s structure to a number. Any conclusion related to the fact that complexity of a system under study can be classified verbally, for example, high, average or low, may be regarded as a linguistic value of complexity conception, and their (values) meanings must be defined more accurately according to certain rules on the basis of the results of system’s structure analysis.

As it is specified by Q-analysis procedure, system’s model represented by a set of orderedpairs or in the equivalent matrix form is replaced by simplices system of simplicial decomposition of complex (polyhedron). Due to this, estimation of structural complexity can be constructed even on the basis of algebraic topology ideas (analysis of simplicial complexes (Y,л) and (X, ), where - adjacent to relation λ). Notation refers to simplicial complex formed by finite set X of simplices and all faces of simplices which are defined by set Y of vertices due to the relation λ. Later on an abridged notations K and can be used for corresponding complexes.

 

3. Make a written translation of the text below.